8. The parabola \(C\) has equation \(y ^ { 2 } = 4 a x\), where \(a\) is a positive constant.
The point \(P \left( a t ^ { 2 } , 2 a t \right)\) lies on \(C\).
- Using calculus, show that the normal to \(C\) at \(P\) has equation
$$y + t x = a t ^ { 3 } + 2 a t$$
The point \(S\) is the focus of the parabola \(C\).
The point \(B\) lies on the positive \(x\)-axis and \(O B = 5 O S\), where \(O\) is the origin. - Write down, in terms of \(a\), the coordinates of the point \(B\).
A circle has centre \(B\) and touches the parabola \(C\) at two distinct points \(Q\) and \(R\).
Given that \(t \neq 0\),
- find the coordinates of the points \(Q\) and \(R\).
- Hence find, in terms of \(a\), the area of triangle \(B Q R\).